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Chapter 5: Problem 38

Find each indefinite integral. $$ \int \frac{x^{2}-1}{x-1} d x $$

Short Answer

Expert verified
The indefinite integral is \( \frac{x^2}{2} + x + C \).

Step by step solution

01

Simplify the integrand

Before integrating, simplify the integrand \( \frac{x^{2}-1}{x-1} \). Notice that the numerator can be factored: \( x^2 - 1 = (x-1)(x+1) \). This allows us to cancel the \( x-1 \) term in the numerator and denominator, simplifying the expression to \( x+1 \).
02

Integrate the simplified expression

Now that the integrand is simplified to \( x+1 \), integrate term by term: \( \int (x+1) \, dx \) becomes \( \int x \, dx + \int 1 \, dx \).
03

Perform the integration

Integrate each term separately. For \( \int x \, dx \), the integral is \( \frac{x^2}{2} \). For \( \int 1 \, dx \), the integral is \( x \). Therefore, the indefinite integral is \( \frac{x^2}{2} + x + C \), where \( C \) is the constant of integration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integrand Simplification
Simplifying the integrand is a crucial first step in solving an indefinite integral. In this problem, you're given the integral \( \int \frac{x^{2}-1}{x-1} \,dx \). The expression under the integral sign is called the integrand, which can often be simplified to make integration easier.
To simplify \( \frac{x^{2}-1}{x-1} \), notice that the numerator is a difference of squares: \( x^2 - 1 = (x-1)(x+1) \). By factoring the numerator, the \( x-1 \) terms in the numerator and denominator cancel each other out, leaving you with the much simpler expression \( x+1 \).
  • Factoring: Look for common factors or patterns such as the difference of squares to break down expressions.
  • Cancelling Out Terms: Simplify fractions by canceling common terms in the numerator and denominator, when possible.
By reducing the integrand to its simplest form, you make the integration process straightforward and more manageable.
Integration Techniques
Integration is the process of finding a function whose derivative is the given function, called the integrand. After simplifying the integrand to \( x+1 \), you can use basic integration techniques to find the indefinite integral.
For each term of the expression, apply the power rule of integration, which states:
  • Power Rule: If \( f(x) = x^{n} \), then \( \int f(x) \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \).
In this problem:
  • Integrating \( x \): Using the power rule, \( \int x \, dx = \frac{x^{2}}{2} \).
  • Integrating a constant (1): For any constant \( a \), \( \int a \, dx = ax \). Therefore, \( \int 1 \, dx = x \).
Combine the results: \( \int (x+1) \, dx = \frac{x^2}{2} + x \). This demonstrates how basic integration techniques simplify the process.
Constant of Integration
When solving indefinite integrals, don't forget the constant of integration, denoted as \( C \). This constant is crucial because indefinite integration represents a family of functions, each differing by a constant.
Observe that:
  • Indefinite integrals lack specific limits or boundaries on the variable of integration. This means the inverse derivative's constant cannot be determined directly.
  • Adding \( C \): After performing integration, always remember to add \( C \). For this problem, \( \int (x+1) \, dx = \frac{x^2}{2} + x + C \).
The constant of integration ensures the most general form of the antiderivative. In application, solving for \( C \) might involve initial conditions or boundary values when dealing with definite integrals. However, in indefinite integrals, \( C \) remains an indispensable symbol of generality.

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